PHYSICAL REVIEW B, VOLUME 64, 193304

Tuning a resonance in Fock space: Optimization of phonon emission in a resonant-tunneling device L. E. F. Foa Torres,1 H. M. Pastawski,1,* and S. S. Makler2,3 1

Facultad de Matema´tica, Astronomı´a y Fı´sica, Universidad Nacional de Co´rdoba, Ciudad Universitaria, 5000 Co´rdoba, Argentina 2 Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro, 21945-970 Rio de Janeiro, Brazil 3 Instituto de Fı´sica, Universidade Federal Fluminense, 24210-340 Nitero´i, Brazil 共Received 9 July 2001; published 12 October 2001兲 Phonon-assisted tunneling in a double-barrier resonant-tunneling device can be seen as a resonance in the electron-phonon Fock space that is tuned by the applied voltage. We show that the geometrical parameters can induce a symmetry condition in this space that can strongly enhance the emission of longitudinal optical phonons. For devices with thin-emitter barriers this is achieved by a wider collector barrier. DOI: 10.1103/PhysRevB.64.193304

PACS number共s兲: 73.40.Gk, 71.38.⫺k, 73.50.Bk, 73.50.Fq

Progress in mesoscopic semiconductor devices1 and molecular electronics2 is driven by the need of miniaturization and the wealth of new physics provided by coherentquantum phenomena. A fundamental idea behind these advances was Landauer’s view that conductance is transmittance.3,4 Hence, the typical conductance peaks and valleys, observed when some control parameter is changed, are seen as fringes in an interferometer. However, the manybody electron-electron (e-e) and electron-phonon (e-ph) interactions restrict the use of this picture. The e-e effects received much attention in different contexts.1 While interest on e⫺ph interaction remained mainly focused on doublebarrier resonant-tunneling devices 共RTD兲,5 where phononassisted tunneling shows up as a satellite peak in a valley of the current-voltage (I-V) curve, recent observation of electromechanical effects in molecular electronics6 requires a reconsideration of the e-ph problem. Theory evolved from a many-body Green’s function in a simplified model for the polaronic states7 to quantum and classical rate equations approach.8 The latter uses an incoherent description of the e-ph interaction by adopting an imaginary self-energy correction to the electronic states.9,10 In this paper, we analyze a quantum coherent solution of transport with e-ph interaction. We resort to a mapping of the many-body problem into a one-body scattering system where each phonon mode adds a new dimension to the electronic variable.11,12 Transmission of electrons between incoming and outgoing channels with different number of phonons are then used in a Landauer’s picture where the only incoherent processes occur inside the electrodes. This allows to develop the concept of resonance in the e-ph Fock space and the identification of the control parameters that optimize the coherent processes leading to a maximized phonon emission. It also gives a clue as to how ‘‘decoherence’’ arise within an exact many-body description. As an application, we consider a RTD phonon emitter where the relevant parameters are best known. There, the first polaronic excitation serves as an ‘‘intermediate’’ state for the phonon emission. An electron with kinetic energy ␧⭐␧ F and potential energy eV in the emitter decays through tunneling into an electron with energy ␧⫹eV⫺ប ␻ 0 in the collector plus a longitudinal optical 共LO兲 phonon. The tuning parameter is the applied voltage while the optimization of phonon emission requires the tailoring of the tunneling rates. 0163-1829/2001/64共19兲/193304共4兲/$20.00

Let us consider a minimal Hamiltonian, H⫽

兺j 兵 E j c †j c j ⫺V j, j⫹1共 c †j c j⫹1 ⫹c †j⫹1 c j 兲 其 ⫹ប ␻ 0 b † b ⫺V g c †0 c 0 共 b † ⫹b 兲 ,

共1兲

where c †j and c j are electron creation and annihilation operators at site j on a 共one-dimensional兲 1D chain with lattice constant a and hopping parameters V j, j⫹1 ⫽V. Tunneling rates are fixed by V 0,1⫽V R and V ⫺1,0⫽V L (V L(R) ⰆV). The site energies are E j ⫽2V for j⬍0 and 2V⫺eV for j⬎0. E 0 ⫽E (o) ⫺ ␣ eV is the well’s ground state 共including the charging effect兲 shifted by the electric field. For barrier widths L L and L R and well size L W a linear approximation for the potential profile gives ␣ ⫽(L L ⫹L W /2)/(L L ⫹L W ⫹L R ). We consider a single LO phonon mode and an interaction V g limited to the well. b † and b are the creation and annihilation operators for phonons. We restrict the Fock space to that expanded by 兩 j,n 典 ⫽c †j (b † ) n / 冑n! 兩 0 典 , which maps to the two-dimensional one-body problem shown in Fig. 1. The number n of phonons is the vertical dimension.11,12 The horizontal dangling chains can be eliminated through a decimation procedure10,13 leading to an effective Hamiltonian,

FIG. 1. Simple model: dots are states in the Fock space, lines are interactions. The effective Hamiltonian including two entangled polaronic states is represented at the bottom.

64 193304-1

©2001 The American Physical Society

BRIEF REPORTS

˜ e-ph⫽ H

兺

n⭓0

PHYSICAL REVIEW B 64 193304

兵 关 E 0 ⫹nប ␻ 0 ⫹⌺ n 共 ␧ 兲兴 兩 0,n 典具 0,n 兩

⫺ 冑n⫹1V g 共 兩 0,n⫹1 典具 0,n 兩 ⫹ 兩 0,n 典具 0,n⫹1 兩 兲 其 . 共2兲 The electron hopping into the electrodes is taken into account by the ␧ dependence of the retarded self-energy corrections ⌺ n ⫽ L ⌺ n ⫹ R ⌺ n . Specifically, L ⌺ n ⫽ 兩 V L /V 兩 2 ⫻⌺(␧⫺nប ␻ 0 ), R ⌺ n ⫽ 兩 V R /V 兩 2 ⫻⌺(␧⫺nប ␻ 0 ⫹eV), with ⌺ 共 ␧ 兲 ⫽⌬ 共 ␧ 兲 ⫺i⌫ 共 ␧ 兲 ;⌬ 共 ␧ 兲 ⫽

1 ␲

冕

⌫共 ␧⬘兲 ␧⫺␧ ⬘

d␧ ⬘ ,

⌫ 共 ␧ 兲 ⫽ 冑V 2 ⫺ 共 ␧/2⫹V 兲 2 ␪ 共 ␧ 兲 ␪ 共 4V⫺␧ 兲 .

共3兲

While the imaginary part ⌫⫽ប v ␧ /a is proportional to the group velocity v ␧ in the electrodes, the actual escape rates ⌫ L(R) /ប are barrier controlled. For width L L(R) and attenuation length ␰ , ⌫ L(R) /⌫⫽ 兩 V L(R) /V 兩 2 ⯝ exp关⫺LL(R) /␰兴. The retarded Green function connecting states i and n, R ˜ e-ph共 ␧ 兲兲 ⫺1 兩 0,i 典 , G n,i 共 ␧ 兲 ⫽ 具 0,n 兩 共 ␧I⫺H

共4兲

has poles at the exact eigenenergies. If ⌺ n (␧)⬅0 , these are the polaronic energies E 0 ⫺ 兩 V g 兩 2 /ប ␻ 0 ⫹nប ␻ 0 . The transmission coefficient T n,i from the ith incoming channel at left electrode to the nth channel at right is10 R T n,i 共 ␧ 兲 ⫽2Im关 R ⌺ n 共 ␧ 兲兴 兩 G n,i 共 ␧ 兲 兩 2 2Im关 L ⌺ i 共 ␧ 兲兴 .

共5兲

If the Fermi energy ␧ F ⰆV, Eq. 共3兲 becomes ⌺ 共 ␧ 兲 ⬇⫺i⌫ 共 ␧⫽␧ F 兲 ␪ 共 ␧ 兲

共6兲

and the ␪ function may cancel some T’s. To obtain the elastic transmittance when g⫽(V g /ប ␻ 0 ) 2 Ⰶ1 and (␧ F ,⌫ L ⫹⌫ R )⬍ប ␻ 0 , we need, R ⯝ G 0,0

1⫺g ¯ 0 ⫹i 关 ⌫ L ⫹⌫ R 兴 ␧⫺E

⫹

⫺ R G 1,0 ⯝

¯ 0 ⫹i 关 ⌫ L ⫹⌫ R 兴 ␧⫺E

, ␧⫺ 关 ¯E 0 ⫹ប ␻ 0 兴 ⫹i 关 ˜⌫ L ⫹⌫ R 兴 共7兲

共8兲

¯ 0. describes the main resonant elastic peak at ␧⫽E The inelastic transmittance T 1,0 can be evaluated from

⫹

Vg ប␻0 ␧⫺ 关 ¯E 0 ⫹ប ␻ 0 兴 ⫹i 关 ˜⌫ L ⫹⌫ R 兴

. 共9兲

When ␧⫹eV⬎ប ␻ 0 escapes are enabled and its poles involve the processes represented in the inset of Fig. 2. 共a兲 The first term gives an inelastic transmittance at the main peak. ¯0 共b兲 The second term provides a satellite peak at ␧⫽E ⫹ប ␻ 0 , associated to a polaronic excitation followed by its decay into an escaping electron and a phonon left behind. Around this satellite peak

g

4⌫ L ⌫ R ⫹O共 g 兲 ¯ 关 ␧⫺E 0 兴 2 ⫹ 关 ⌫ L ⫹⌫ R 兴 2

Vg ប␻0

T 1,0⯝

evaluated with the first two polaronic states. Here, ˜⌫ L ⫽g⌫ L and ¯E 0 ⫽E 0 ⫺ 兩 V g 兩 2 /ប ␻ 0 . The first term contains the main resonance associated to the build up of the polaronic ground state. The second term contains a virtual exploration into the first polaronic excitation. It is noteworthy that when ⌫⫽0, this Green function would cancel out at an intermediate energy giving rise to an antiresonance.14,13 This concept extends the spectroscopic Fano resonances15 to the problem of conductance.16 For gⰆ1, this effect is less important and in the whole energy range, T 0,0⯝

FIG. 2. Current density as a function of the applied voltage for a symmetrical 共thin line兲 and optimized 共thick line兲 structures with L L ⫽7a (19.7 Å兲. The dotted line indicates the background current in the region of the satellite peak for the symmetrical structure. The inelastic processes contributing to the peaks 共a兲 and 共b兲 are represented in the inset.

˜ L⌫ R 4⌫ E 0 ⫹ប ␻ 0 兲兴 2 ⫹ 关 ˜⌫ L ⫹⌫ R 兴 2 关 ␧⫺ 共 ¯

,

共10兲

showing that phonon emission is a resonance in the Fockspace 共see bottom of Fig. 1兲. A maximal probability (T 1,0 ⫽1) requires equal rates of formation and decay:14 ˜⌫ L ⫽⌫ R , which in our RTD implies

冋 册

L R ⯝L L ⫹2 ␰ ln

ប␻o . Vg

共11兲

Hence, thin barriers with this generalized symmetry condition have T 1,0⯝1 over a broad energy range. The application of the Keldysh formalism17 to our Fock space gives an electrical current Itot expressed as a balance equation4 in terms of the transmittances of Eq. 共5兲 and the electrochemical potentials. The experimental condition of high bias and low temperature (eV⬎␧ F Ⰷk B T), rules out right-to-left flow, while ប ␻ o ⬎␧ F , enables the ␪ in Eq. 共6兲 preventing inelastic reflection and overflow18 of the final states. Thus,

193304-2

BRIEF REPORTS

I tot⫽

PHYSICAL REVIEW B 64 193304

兺n I n ;

where I n ⫽

冉 冊冕 2e h

␧F

0

T n,0共 ␧ 兲 d␧.

共12兲

The ‘‘decoherence’’ introduced by the e-ph interaction on the former single-particle description can be now appreciated. One aspect, valid even if ប ␻ o →0, is that in Eq. 共12兲 the outgoing currents cannot interfere. Another is the phaseshift fluctuations and ‘‘broadening’’ of the one-particle resonant energy induced by the virtual processes in the elastic channel of Eq. 共7兲. At the satellite peak, the main elastic contribution to the current is provided by the off-resonant tunneling through the ground state, i.e., I 0 ⯝2e/h4⌫ L ⌫ R ␧ F /(ប ␻ 0 ) 2 . The inelastic current determined by Eqs. 共10兲 and 共12兲 is I 1⯝

⯝

冋 冉

˜ L⌫ R 2 ␧F e 4⌫ ⫻ arctan ប 共 ˜⌫ L ⫹⌫ R 兲 ␲ 2 共 ˜⌫ L ⫹⌫ R 兲

再

冊册

e ˜ ⌫ / 共 ˜⌫ L ⫹⌫ R 兲 4⌫ ប L R

for ␧ F Ⰷ 共 ˜⌫ L ⫹⌫ R 兲

2e T ⫻␧ h 1,0 F

for ␧ F Ⰶ 共 ˜⌫ L ⫹⌫ R 兲

FIG. 3. Power emitted as LO phonons as a function of the applied voltage for L L ⫽7a (19.7 Å兲 and different values of L R . The efficiency as a function of the right barrier width is shown in the inset.

.

共13兲

The first line differs from the result of rate equations in Ref. 8 by the factor in brackets, fundamental to resolve extreme ˜ L ⫹⌫ R ) the inelastic current becomes regimes. When ␧ F Ⰷ(⌫ ˜ L . In geometry independent in the wide range of ␧ F Ⰷ⌫ R ⬎⌫ the opposite case I 1 , and hence the power emitted as phonons ប ␻ 0 I 1 /e, becomes determined by the transmittance at resonance, which is maximized by the generalized symmetry condition of Eq. 共11兲. Each current term, I n⬎0 , contributes with n useful phonons, while I 0 ’s energy degrades fully into electrode heating. Then, one might seek a maximal ratio between the inelastic power P in and the total power P,

␩⫽

P in ⫽ P

ប␻0

I

n n 兺 e n⬎0

I totV

,

共14兲

which is the efficiency to transform the electric potential energy into LO phonon energy. At the voltage tuning the resonance at the satellite peak V 0 ⯝(E (o) ⫺ 兩 V g 兩 2 /ប ␻ 0 ⫹ប ␻ 0 ⫺␧ F /2)/ ␣ , the lowest order of ␩ has two factors I 1 /(I 0 ⫹I 1 ) and ប ␻ 0 /eV 0 . The first is small for narrow barriers because nonresonant tunneling dominates over phononassisted tunneling. For wide barriers, it goes to one as ˜⌫ L ⫹⌫ R →0 . The second decreases with increasing right barrier’s width because it requires a higher V 0 . Thus, as ⌫ R is decreased, two effects compete: the switch from nonresonant to phonon-assisted resonant tunneling and an excess in the electronic kinetic energy in the collector. Hence, as long as the left barrier is not extremely thin (⌫ L⬎ប ␻ 0 ), ␩ can not depend much on geometry. With this restriction in mind, a device designed for phonon production should maximize the emitted power according to Eq. 共11兲. Let us compare these basic predictions with the numerical results of a description involving geometry, voltage, and en-

ergy dependences of a typical RTD. A discrete 3D model is defined in terms of the effective mass m * with V ⫽ប 2 /(2m * a 2 ). The potential profile for the diagonal energies E j is shown in the inset of Fig. 2. N L , N R and N w are the number of sites in the left and right barriers and the well, the associated widths are L i ⫽N i a. For translational symmetry along the interface, we consider a single phonon mode per transversal 共parallel to the interface兲 state, with frequency ␻ 0 and localized in the structure region. While conservation of transverse electron’s momentum might not be fully realistic,19 it constitutes a first approximation yielding results consistent with the main experimental features. The current components are obtained from Eq. 共12兲 by integration over the transversal modes. The parameters in our calculations are chosen to simulate the case of a GaAs-Al1⫺x Gax As structure. The effective mass m * is 0.067m e , the LO phonon frequency ប ␻ 0 ⫽36 meV, a ⫽2.825 Å, and the hopping parameter V⫽7.125 eV. A typical e-ph interaction strength of g⬃0.1 is obtained with V g ⯝10 meV. For a well of 56.5 Å , barrier heights of 300 meV and ␧ F ⫽10 meV, the inclusion of n⭐3 warrants good numerical convergence. For wide left barriers 共of about 25a⯝70 Å or more兲, we found that the maximum value of P in varies slowly with the width of the right barrier. Hence, consistently with our discussion of the 1D model, there is no substantial gain in P in by choosing an asymmetric structure. Consequently, a high phonon emission rate should be sought for thin barriers. A thin left barrier of L L⫽7a 共19.7 Å兲, gives a tunneling probability T L (␧ F )⫽⌫ L /⌫⬃0.03. Figure 2 shows the I⫺V curves for symmetric and asymmetric RTD’s. In Fig. 3 we show P in ⫺V for various right barrier widths L R . The peaks are shifted to higher voltages as L R is increased, because the resonant energies are lowered approximately by ␣ eV. We can also see that the peak value of P in as a function of the right barrier width exhibits a maximum. The I-V curve for the optimal configuration is shown in Fig. 2 共heavy line兲. The inset of Fig. 3 shows the dependence of ␩ , evaluated at the

193304-3

BRIEF REPORTS

PHYSICAL REVIEW B 64 193304

optimal voltage, on the right barrier width for various left barrier widths where ⌫ L ⬍ប ␻ 0 . In agreement with our theoretical analysis, ␩ keeps the same magnitude for all the shown geometrical configurations. The main result of Fig. 3 is the confirmation that, for a given L L satisfying ⌫ L ⬍ប ␻ 0 and g⌫ L ⫹⌫ R ⬎␧ F , the phonon emission rate is enhanced by a factor 2.5 by choosing a wider right barrier as prescribed by Eq. 共11兲. This may explain the unusually large satellite peaks of asymmetric structures.19 An RTD optimized for phonon emission might have many applications. In fact, in Al1⫺x Gax As-GaAs RTD these primary LO phonons have a short life time20 and decay into a pair of LO and transverse acoustic 共TA兲 phonons. This phenomenon inspired the proposal11 for the generation of a coherent TA-phonon beam in an RTD 共called a SASER兲.21 That device required an energy difference between the first two electronic states in the well E 1 ⫺E 0 ⫽ប ␻ 0 .11,21 In contrast,

*Corresponding author Email address: [email protected] 1

L. P. Kouwenhoven et al., in Mesoscopic Electron Transport, edited by L. L. Sohn, L. P. Kouwenhoven, and G. Scho¨n, 共Kluwer, Dordrecht, 1997兲. 2 C. Joachim, J. K. Gimzewski, and A. Aviram, Nature 共London兲 408, 541 共2000兲. 3 R. Landauer, Phys. Lett. 85A, 91 共1981兲. 4 M. Bu¨ttiker, Phys. Rev. Lett. 57, 1761 共1986兲. 5 V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys. Rev. B 36, 7635 共1987兲; M. L. Leadbeater et al., ibid. 39, 3438 共1989兲; G. S. Boebinger et al., Phys. Rev. Lett. 65, 235 共1990兲. 6 L. P. Kouwenhoven, Nature 共London兲 407, 35 共2000兲; H. Park, et al., 407, 57 共2000兲; B. C. Stipe, M. A. Rezaei, and W. Ho, Phys. Rev. Lett. 81, 1263 共1998兲. 7 N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins, Phys. Rev. Lett. 61, 1396 共1988兲. 8 R. G. Lake, G. Klimeck, M. P. Anantram, and S. Datta, Phys. Rev. B 48, 15132 共1993兲. 9 S. Datta, Phys. Rev. B 40, 5830 共1989兲. 10 J. L. D’Amato and H. M. Pastawski, Phys. Rev. B 41, 7411 共1990兲; see also H. M. Pastawski and E. Medina, Rev. Mex. Fis.

the present proposal does not require such an accurate device geometry. Instead, operation in the phonon emission mode only requires the tuning of the many-body resonance with the external voltage. Geometry just improves its yield by imposing a generalized symmetry condition in the Fock space. For a typical Al1⫺x Gax As emitter barrier of 20 Å this would require a 54 Å collector’s barrier. We expect that our results could stimulate the search for excited phonon modes 共e.g. with Raman spectroscopy兲, in operational RTD’s as a function of the applied voltage in the various configurations. While for simplicity we have restricted our analysis to a model RTD, our analysis applies to other problems6 involving electronic resonant tunneling in the presence of an interaction with an elementary excitation. We acknowledge financial support from CONICET, SeCyT-UNC, ANPCyT, and Andes-Vitae-Antorchas. H.M.P. and L.E.F.F.T. are affiliated with CONICET.

47S1, 1 共2001兲 available at cond-mat/0103219. E. V. Anda, S. S. Makler, H. M. Pastawski, and R. G. Barrera, Braz. J. Phys. 24, 330 共1994兲. 12 J. Boncˇa and S. A. Trugman, Phys. Rev. Lett. 75, 2566 共1995兲; 79, 4874 共1997兲. 13 P. Levstein, H. M. Pastawski, and J. L. D ’Amato, J. Phys.: Condens. Matter 2, 1781 共1990兲. 14 J. L. D’Amato, H. M. Pastawski, and J. F. Weisz, Phys. Rev. B 39, 3554 共1989兲. 15 U. Fano, Phys. Rev. 124, 1866 共1961兲. 16 J. Go¨res et al., Phys. Rev. B 62, 2188 共2000兲. 17 H. M. Pastawski, Phys. Rev. B 46, 4053 共1992兲. 18 E. G. Emberly and G. Kirczenow, Phys. Rev. B 61, 5740 共2000兲. 19 P. J. Turley, C. R. Wallis, S. W. Teitsworth, W. Li, and P. K. Bhattacharya, Phys. Rev. B 47, 12640 共1993兲. 20 F. Valle´e, Phys. Rev. B 49, 2460 共1994兲. 21 S. S. Makler, M. I. Vasilevskiy, E. V. Anda, D. E. Tuyarot, J. Weberszpil, and H. M. Pastawski, J. Phys. Condens. Matter 10, 5905 共1998兲; I. Camps and S. S. Makler, Solid State Commun. 116, 191 共2000兲. 11

193304-4